Concerning the Harnack type inequality, in [1]}, Trudinger established Harnack and weak Harnack type estimates for weak solutions to general second order quasilinear problems. As an application to these, the author obtains Hölder continuity for weak solutions.
However, in the nonlocal case, the classical Harnack inequality fails, see for instance [2]}. Subsequently,
in [3]}, the same author proved the Harnack inequality (where the nonnegativity in the whole \(\mathbb {R}^N\)  is not assumed) with nonlocal tails for problems involving the fractional Laplacian. For general \(p\) , we refer to [4]} for Harnack and weak Harnack type inequality for minimizers as well as weak solutions. Here, authors have used suitable Caccioppoli type estimates and a local boundedness result to achieve these aims.
